This is my first time doing an ongoing math circle with many sessions devoted to one topic. It's also my first time getting my own students to come to a math circle. I am really happy that they keep coming. I originally said it would be five sessions, but I can see that we could easily go for six to eight sessions on the questions raised here. (I may let them talk me into extending it.)

I love it when the way they approach a problem is different than the way I would have done it. X saw a pattern I had not seen before, and we explored her pattern at length in the second session. I haven't had time to write up the details, and have probably forgotten much of it.

**Week One**

What examples can we come up with? (3-4-5, 5-12-13, ...)

6-8-10 leads us to define primitive Pythagorean triples (in which gcf(a,b,c)=1; 6-8-10 isn't primitive)

Maybe writing a list of all the perfect squares up to 400 will help us find more.

What patterns do we see?

- Odd + Even = Odd
- Middle number is a multiple of 4
- c = b+1 (after which I added 8-15-17 to our list)

**Week Two**

One person was new, so we reviewed our first week's work for him.

We explored the "family" of triples with c = b+1. a

^{2}+ b

^{2}= (b+1)

^{2}becomes a

^{2}= (b+1)

^{2}- b

^{2}

= b

^{2}+ 2b + 1 - b

^{2}= 2b-1. If a

^{2}= 2b-1, then b = (a

^{2}+1)/2. This will be a whole number whenever a is an odd number. So we got lots more: 7-24-25, 9-40-41, ...

X noticed that in the triples

3-4-5the second number is 4*1, 4*3, 4*6, 4*10, ... For the nth one, we use 4 times a number n more than the previous one. I showed them why these (1, 3, 6, 10, ...) are called triangle numbers, and asked them to add 1 to 100. They each came up with their own way of thinking about it. We came back to X's pattern and wrote:

5-12-13

7-24-25

9-40-41

a=2n+1

b=4*n(n+1)/2=2n(n+1)

c=b+1

**Week Three**

Another new person came, so we summarized for her. Then we explored triples where c = b+2.

I love seeing their creativity and persistence. At the same time, I am blown away by the holes in their understanding of algebra moves. Y was considering (4n)

^{2}, and thought he might have to distribute.

We verified that we get all of the primitive Pythagorean triples with c=b+2 using:

a=4n

b=4n^{2}-1

c=4^{2}+1

Not sure where we'll take it in Week Four, but eager to find out. I am still struggling to lead less, become less visible, and listen more.

Love it, Sue! Keep us posted!

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